Paulo C. Rech

Hopfield neural network: The hyperbolic tangent and the piecewise-linear activation functions

This paper reports two-dimensional parameter-space plots for both, the hyperbolic tangent and the piecewise-linear neuron activation functions of a three-dimensional Hopfield neural network. The plots obtained using both neuron activation functions are compared, and we show that similar features are present on them. The occurrence of self-organized periodic structures embedded in chaotic regions is verified for the two cases.

A HYPERCHAOTIC CHUA SYSTEM

In this paper, we report a new four-dimensional autonomous hyperchaotic system, constructed from a Chua system where the piecewise-linear function usually taken to describe the nonlinearity of the Chua diode has been replaced by a cubic polynomial. Analytical and numerical procedures are conducted to study the dynamical behavior of the proposed new hyperchaotic system.

Spiral periodic structure inside chaotic region in parameter-space of a Chua circuit

In this letter we investigate, via numerical simulations, the parameter-space of the set of autonomous first-order differential equations of a Chua circuit. We show that this parameter-space presents self-organized periodic structures immersed in a chaotic region, forming a single spiral structure that coils up around a focal point. Additionally, bifurcation diagrams are used to show that those periodic structures also organize themselves in period-adding cascades, along specific directions that point towards this same focal point. Copyright

Some two-dimensional parameter spaces of a Chua system with cubic nonlinearity

In this paper we investigate three two-dimensional parameter spaces of a three-parameter set of autonomous differential equations used to model the Chua oscillator, where the piecewise-linear function usually taken to describe the nonlinearity of the Chua diode has been replaced by a cubic polynomial. It is made by using three independent two-dimensional cross sections of the three-dimensional parameter space generated by the model, which contains three parameters. We show that, independent of the parameter set considered in plots, all the diagrams present periodic structures embedded in a large chaotic region, and we also show that these structures organize themselves in period-adding cascades. We argue that these selected two-dimensional cross sections can be representative of the three-dimensional parameter space as a whole, in the range of parameters here investigated.

Dynamics in the Parameter Space of a Neuron Model

Some two-dimensional parameter-space diagrams are numerically obtained by considering the largest Lyapunov exponent for a four-dimensional thirteen-parameter Hindmarsh—Rose neuron model. Several different parameter planes are considered, and it is shown that depending on the combination of parameters, a typical scenario can be preserved: for some choice of two parameters, the parameter plane presents a comb-shaped chaotic region embedded in a large periodic region. It is also shown that there exist regions close to these comb-shaped chaotic regions, separated by the comb teeth, organizing themselves in period-adding bifurcation cascades.

A PARAMETER-SPACE OF A CHUA SYSTEM WITH A SMOOTH NONLINEARITY

In this paper we investigate, via numerical simulations, the parameter space of the set of autonomous differential equations of a Chua oscillator, where the piecewise-linear function usually taken to describe the nonlinearity of the Chua diode was replaced by a cubic polynomial. As far as we know, we are the first to report that this parameter-space presents islands of periodicity embedded in a sea of chaos, scenario typically observed only in discrete-time models until recently. We show that these islands are self-similar, and organize themselves in period-adding bifurcation cascades.

Period-adding and spiral organization of the periodicity in a Hopfield neural network

This work reports two-dimensional parameter space plots, concerned with a three-dimensional Hopfield-type neural network with a hyperbolic tangent as the activation function. It shows that typical periodic structures embedded in a chaotic region, called shrimps, organize themselves in two independent ways: (i) as spirals that individually coil up toward a focal point while undergo period-adding bifurcations and, (ii) as a sequence with a well-defined law of formation, constituted by two different period-adding sequences inserted between.

Lyapunov exponent diagrams of a 4-dimensional Chua system

We report numerical results on the existence of periodic structures embedded in chaotic and hyperchaotic regions on the Lyapunov exponent diagrams of a 4-dimensional Chua system. The model was obtained from the 3-dimensional Chua system by the introduction of a feedback controller. Both the largest and the second largest Lyapunov exponents were considered in our colorful Lyapunov exponent diagrams, and allowed us to characterize periodic structures and regions of chaos and hyperchaos. The shrimp-shaped periodic structures appear to be malformed on some of Lyapunov exponent diagrams, and they present two different bifurcation scenarios to chaos when passing the boundaries of itself, namely via period-doubling and crisis. Hyperchaos-chaos transition can also be observed on the Lyapunov exponent diagrams for the second largest exponent.

Hyperchaotic states in the parameter-space

Abstract In this paper we propose a numerical method to characterize hyperchaotic points in the parameter-space of continuous-time dynamical systems. The method considers the second largest Lyapunov exponent value as a measure of hyperchaotic motion, to construct two-dimensional parameter-space color plots. Different levels of hyperchaos in these plots are represented by a continuously changing yellow–red scale. As an example, a particular system modeled by a set of four nonlinear autonomous first-order ordinary differential equations is considered. Practical applications of these plots include, by instance, walking in the parameter-space of hyperchaotic systems along desirable paths.

Period-Adding Structures in the Parameter-Space of a Driven Josephson Junction

Two-dimensional parameter-space diagrams related to a driven Josephson junction are reported. Three cases are considered, namely those involving the external direct current as one of the parameters. Typical periodic structures embedded in a chaotic region are observed in all diagrams, organized in different ways: (i) As structures with a similar shape to the Arnold tongues of the circle map, in period-adding sequences, and (ii) as structures with other shapes, in arrangements including two mixed sets of period-adding sequences.